Journal of Theoretical Probability

, Volume 23, Issue 3, pp 888–903 | Cite as

Martingale Approximation and Optimality of Some Conditions for the Central Limit Theorem

  • Dalibor Volný


Let (X i ) be a stationary and ergodic Markov chain with kernel Q and f an L 2 function on its state space. If Q is a normal operator and f=(IQ)1/2 g (which is equivalent to the convergence of \(\sum_{n=1}^{\infty}\frac{\sum_{k=0}^{n-1}Q^{k}f}{n^{3/2}}\) in L 2), we have the central limit theorem [cf. (Derriennic and Lin in C.R. Acad. Sci. Paris, Sér. I 323:1053–1057, 1996; Gordin and Lifšic in Third Vilnius conference on probability and statistics, vol. 1, pp. 147–148, 1981)]. Without assuming normality of Q, the CLT is implied by the convergence of \(\sum_{n=1}^{\infty}\frac{\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}}{n^{3/2}}\) , in particular by \(\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}=o(\sqrt{n}/\log^{q}n)\) , q>1 by Maxwell and Woodroofe (Ann. Probab. 28:713–724, 2000) and Wu and Woodroofe (Ann. Probab. 32:1674–1690, 2004), respectively. We show that if Q is not normal and f∈(IQ)1/2 L 2, or if the conditions of Maxwell and Woodroofe or of Wu and Woodroofe are weakened to \(\sum_{n=1}^{\infty}c_{n}\frac{\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}}{n^{3/2}}<\infty\) for some sequence c n ↘0, or by \(\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}=O(\sqrt{n}/\log n)\) , the CLT need not hold.


Martingale approximation Martingale difference sequence Strictly stationary process Markov chain Central limit theorem 

Mathematics Subject Classification (2000)

60G10 60G42 28D05 60F05 


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  1. 1.
    Cornfeld, I.P., Fomin, S.V., Sinai, Y.G.: Ergodic Theory. Springer, Berlin (1982) MATHGoogle Scholar
  2. 2.
    Cuny, C.: Pointwise ergodic theorems with rate and application to limit theorems for stationary processes. arXiv:0904.0185 (2009, submitted for publication)
  3. 3.
    Cuny, C.: Norm convergence of some power-series of operators in L p with applications in ergodic theory (2009, submitted for publication) Google Scholar
  4. 4.
    Derriennic, Y., Lin, M.: Sur le théorème limite central de Kipnis et Varadhan pour les chaînes réversibles ou normales. C.R. Acad. Sci. Paris, Sér. I 323, 1053–1057 (1996) MATHMathSciNetGoogle Scholar
  5. 5.
    Derriennic, Y., Lin, M.: The central limit theorem for Markov chains with normal transition operators, started at a point. Probab. Theory Relat. Fields 119, 509–528 (2001) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Gordin, M.I.: A central limit theorem for stationary processes. Sov. Math., Dokl. 10, 1174–1176 (1969) MATHGoogle Scholar
  7. 7.
    Gordin, M.I., Holzmann, H.: The central limit theorem for stationary Markov chains under invariant splittings. Stoch. Dyn. 4, 15–30 (2004) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gordin, M.I., Lifšic, B.A.: Central limit theorem for stationary processes. Sov. Math., Dokl. 19, 392–394 (1978) MATHGoogle Scholar
  9. 9.
    Gordin, M.I., Lifšic, B.A.: A remark about a Markov process with normal transition operator. In: Third Vilnius Conference on Probability and Statistics, vol. 1, pp. 147–148 (1981) Google Scholar
  10. 10.
    Gordin, M.I., Lifšic, B.A.: The central limit theorem for Markov processes with normal transition operator, and a strong form of the central limit theorem. In: Borodin, A., Ibragimov, I. (Eds.) Limit Theorems for Functionals of Random Walks, Proc. Steklov Inst. Math., vol. 195. Am. Math. Soc., Providence (1994). Sects. IV.7 and IV.8, English Translation Am. Math. Soc., Providence (1995) Google Scholar
  11. 11.
    Kipnis, C., Varadhan, S.R.S.: Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun. Math. Phys. 104, 1–19 (1986) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Klicnarová, J., Volný, D.: Exactness of a Wu–Woodroofe’s approximation with linear growth of variances. Stoch. Process. Their Appl. 119, 2158–2165 (2009) MATHCrossRefGoogle Scholar
  13. 13.
    Maxwell, M., Woodroofe, M.: Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28, 713–724 (2000) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Peligrad, M., Utev, S.: A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33, 798–815 (2005) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Rosenblatt, M.: Markov Processes: Structure and Asymptotic Behavior. Springer, Berlin (1971) MATHGoogle Scholar
  16. 16.
    Volný, D.: Approximating martingales and the central limit theorem for strictly stationary processes. Stoch. Process. Their Appl. 44, 41–74 (1993) MATHCrossRefGoogle Scholar
  17. 17.
    Volný, D.: Martingale approximation of non-adapted stochastic processes with nonlinear growth of variance. In: Bertail, P., Doukhan, P., Soulier, P. (Eds.) Dependence in Probability and Statistics. Lecture Notes in Statistics, vol. 187, pp. 141–156. Springer, New York (2006) CrossRefGoogle Scholar
  18. 18.
    Wu, W.B., Woodroofe, M.: Martingale approximation for sums of stationary processes. Ann. Probab. 32, 1674–1690 (2004) MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de RouenSaint Etienne du RouvrayFrance

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